Rigorous spectral analysis of the metal-insulator transition in a limit-periodic potential (Q1284925)

One of the possible mechanisms leading to a metal-insulator transition with a one-particle theory is based upon quantum effect in almost-periodic systems. This transition is expected to be continuous: as the strength of the almost periodic potential increases, the Hausdorff dimension of the singular continuous local density of states should decrease from 1, the Hausdorff dimension of an absolutely continuous measure, towards 0. The purpose of this paper is to exhibit an example for which the above scheme can be proved to be correct. To this end, the authors consider the limit-periodic Jacobi matrices associated to the real Julia set \(f_\lambda(z)= z^2-\lambda\), \(z\in\mathbb{C}\) and \(\lambda\in [2,\infty)\). The typical local spectral exponent of their spectral measures is shown to be a harmonic function in \(\lambda\) decreasing logarithmically from 1 to 0.